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APPENDIX A                                 339
(e)   •coB,.g-g + g-g+...>[01-1]
= 1.   [When z<O, and consequently xz, xz, etc.,
(a)      eix = cos x + i sin x.
(b)      e-"** = cos x — i sin x.
1      2!     3!     4!            }
= x.    [When x <C 1, and consequently x2, s3, etc.,
VII. RELATIONS WHICH  CONNECT  EXPONENTIAL FUNCTIONS WITH CIRCULAR FUNCTIONS.
"These are called De Moivre's Theorems" and are obtained by comparing series (b) and (c) of VI with series (d) and (e) of _the same group.
(c)     sin x = &iX ~~ e~~iX~-      r™s Cation is obtained by subtract-2i            [ing (b) from (a).
/ IN              __ eix + e-ix      fThis relation is obtained by adding (b)"
^QJ        COS X —            ~           •        I ,     /  \
2              [to (a).
VIII. HYPERBOLIC FUNCTIONS.
(a)      sinh x = i sin (ix).
(b)      coshrc = cos (ix).
These are the definitions of the hyperbolic sine and the hyperbolic cosine. Replacing x by ix in equations (c) and (d) of group VII we obtain the following relations between hyperbolic and exponential functions:
Qx _ e—x
(c)      sinh x = —r----,
z
(d)      cosh x = e '•    e— -
Squaring equation (c) and subtracting it from the square of equation (d) we obtain
(e)     cosh2 x — sinh2 x = 1.
IX. AVERAGE VALUE.
The average value of y == / (x) in the interval between x = x\ and x = xz is given by
£2 —